Higher order Kantorovich-type Szász–Mirakjan operators

In the present paper, we introduce the genuine q-Stancu-Bernstein–Durrmeyer operators Znq,α(f;x). We calculate the moments of these operators, Znq,α(tj;x) for j=0,1,2, which follows a symmetric pattern. We also calculate the second order central moment Znq,α((t−x)2;x). We give a Korovkin-type theorem; we estimate the rate of convergence for continuous functions. Furthermore, we prove a local approximation theorem in terms of second modulus of continuity; we obtain a local direct estimate for the genuine q-Stancu-Bernstein–Durrmeyer operators in terms of Lipschitz-type maximal function of order β and we prove a direct global approximation theorem by using the Ditzian-Totik modulus of second order.

The paper aims at approximating functions through a sequence of linear positive operators of continuous type. First we define the Appell polynomials of dimension d(d≥2) and with their aid we expand the Szász–Mirakjan operators in the Durrmeyer sense. The investigation of the new class of operators involves the study of convergence by using the universal Bohman–Korovkin theorem and the indication of the error by using modulus of smoothness. The main results are given by the asymptotic expansion of both the operators and their derivatives with the explicit specification of all coefficients in a concise form. In particular, Voronovskaja type theorems are obtained.

This paper is interested in a new sequence of linear positive operators including degenerate Appell polynomials. We give a convergence theorem for these operators and obtain the quantitative estimation of the approximation by using modulus of continuity, Peetre’s 𝒦-functional, Lipschitz class functions and a Voronovskaja-type theorem. In addition, we give a Kantorovich modification of these operators and derive some approximation properties.

The main object of this paper is to construct a new class of modified (p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(p,q)$\\end{document}-Gamma-type operators. For this new class of operators, in section one, the general moments are find; in section two, the Korovkin-type theorem and some direct results are proved by considering the modulus of continuity and modulus of smoothness and their behavior in Lipschitz-type spaces. In section three, some results in the weighted spaces are given, and in the end, some shape-preserving properties are proven.

The concept of f-lacunary statistical convergence which is, in fact, a generalization of lacunary statistical convergence, has been introduced recently by Bhardwaj and Dhawan (Abstr. Appl. Anal. 2016:9365037, 2016). The main object of this paper is to prove Korovkin type approximation theorems using the notion of f-lacunary statistical convergence. A relationship between the newly established Korovkin type approximation theorems via f-lacunary statistical convergence, the classical Korovkin theorems and their lacunary statistical analogs has been studied. A new concept of f-lacunary statistical convergence of degree β (0 < beta< 1) has also been introduced, and as an application a corresponding Korovkin type theorem is established.

This article is a continuation of our previous works. We mainly investigate a Korovkin type theorem for double sequences of positive linear operators defined in the space of all $2\pi $-periodic and real valued continuous functions on the real two-dimensional space with help of the concept of triangular $A$-statistical convergence, which is a kind of statistical convergence for double real sequences. Also, we analyze the rate of convergence of double operators in this via modulus of continuity.

In this note, we study approximation properties of a family of linear positive operators and establish asymptotic formula, rate of convergence, local approximation theorem, global approximation theorem, weighted approximation theorem, and better approximation for this family of linear positive operators.

In this article, we present the Durrmeyer variant of generalized Bernstein operators that preserve the constant functions involving a non-negative parameter $$\rho $$ . We derive the approximation behaviour of these operators including a global approximation theorem via Ditzian–Totik modulus of continuity and the order of convergence for the Lipschitz type space. Furthermore, we study a Voronovskaja type asymptotic formula, local approximation theorem by means of second order modulus of smoothness and the rate of approximation for absolutely continuous functions having a derivative equivalent to a function of bounded variation. Lastly, we illustrate the convergence of these operators for certain functions using Maple software.

In the present paper, we introduce q-analogue of the Jakimovski-Leviatan operators with the help of q-Appell polynomials. We establish some moments and auxiliary results by using q-derivatives and then prove a basic convergence theorem. Also, the Voronovskaja-type asymptotic formula and some direct results for the above operators are discussed. Moreover, the rate of convergence and weighted approximation by these operators in terms of modulus of continuity are studied.

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We use the new variant of Szász–Mirakjan operators to construct a generalized version of Szász‐beta type operators and obtain auxiliary lemmas. We present the weighted approximation theorems and, by using Peetre’s K ‐function, the local approximation results of these operators are studied. We also analyze the rate of convergence by utilizing Lipschitz‐type maximal functions and the ordinary modulus of continuity. Moreover, using aforesaid operators, we finally examine the Voronovskaya‐type theorem.

Weighted statistical convergence based on generalized difference operator involving (p,q)-gamma function and its applications to approximation theorems

In this paper, a generalized sequence of Baskakov operators connected to the Riemann–Liouville fractional integral is introduced. These sequences of operators deal with smooth approximation behavior in a wider class, i.e., a class of measurable functions in the Lebesgue sense. Further, estimates in terms of test functions and central moments are established. Using the weighted Korovkin-type theorem, uniform convergence in weighted spaces is studied. Additionally, uses of the classical modulus of continuity and Peetre’s K-functional to establish local and global direct approximation theorems are introduced. Lastly, the statistical convergence properties of the suggested operators are investigated. In the last section, theoretical results are supported by numerical examples with graphical and numerical illustrations.

In this paper, we introduce a new type Kantorovich variant of Sz?sz-Mirakjan operators based on q-integers. We study convergence properties by using Korovkin?s theorem and estimate the rate of convergence by using modulus of continuity. We examine local and weighted approximation properties in terms of modulus of continuity. We obtain a direct estimate in terms of Lipschitz type maximal function of order ?. Moreover, we give a quantitative Voronovskaja type theorem for these newly defined operators. Finally, we present numerical results and graphical representation.

We introduce Kantorovich variant of Stancu-Lupas? operators and study convergence and qstatistical convergence properties using Korovkin theorem. Rate of convergence is analyzed in terms of modulus of continuity, elements of Lipschitz class and Peetre?s K-functional. Direct theorems are proved and Voronovskaja type theorem is established. Graphical analysis of convergence and error estimations are presented with the help of MATLAB.